Solve for $x$ : $ 4|x + 5| + 8 = 5|x + 5| + 4 $
Explanation: Subtract $ {4|x + 5|} $ from both sides: $ \begin{eqnarray} 4|x + 5| + 8 &=& 5|x + 5| + 4 \\ \\ {- 4|x + 5|} && {- 4|x + 5|} \\ \\ 8 &=& 1|x + 5| + 4 \end{eqnarray} $ Subtract $4$ from both sides: $ \begin{eqnarray} 8 &=& 1|x + 5| + 4 \\ \\ {- 4} && {- 4} \\ \\ 4 &=& 1|x + 5| \end{eqnarray} $ Simplify: $ 4 = |x + 5| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -4 = x + 5 $ or $ 4 = x + 5 $ Solve for the solution where $x + 5$ is negative: $ - 4 = x + 5$ Subtract ${5}$ from both sides: $ \begin{eqnarray} - 4 &=& x + 5 \\ \\ {- 5} && {- 5} \\ \\ -4 - 5 &=& x \end{eqnarray} $ $ -9 = x $ Then calculate the solution where $x + 5$ is positive: $ 4 = x + 5 $ Subtract ${5}$ from both sides: $ \begin{eqnarray} 4 &=& x + 5 \\ \\ {- 5} && {- 5} \\ \\ 4 - 5 &=& x \end{eqnarray} $ $ -1 = x $ Thus, the correct answer is $x = -9 $ or $x = -1 $.